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Full text of "Scientific Papers - Vi"

1914]                  ON  THE STABILITY  OF  VISCOUS  FLUID  MOTION                     269
"In your p. 62, lines 11 and 12 are 'inaccurate.5 Stokes limits his investigation to the case in which the squares of the velocities can be neglected
,.    radius of globe x velocity               11X
(i.e.---------,. £••. .,------------- very small).
v               diifusivity                    J          "
in which it is manifest that the steady motion is the same whatever the viscosity; but it is manifest that when the squares cannot be neglected, the steady motion is very different (and horribly difficult to find) for different degrees of viscosity.
"In your p. 62, near the foot, it is not explained what V is; and it disappears henceforth.—Great want of explanation here—Did you not want your paper to be understandable without Basset in hand ? I find your two papers of July/92, pp. 61—70, andOct./93,pp. 355—372, very difficult reading, in every page, and in some oc ly difficult.
" Pp. 366, 367 very mysterious. The elastic problem is not defined. It is impossible that there can be the rectilineal motion of the fluid asserted in p. 367, lines 17—19 from foot, in circumstances of motion, quite undefined, but of some kind making the lines of motion on the right side different from those on the left. The conditions are not explained for either the elastic-solid*, or the hydraulic case.
" See p. 361, lines 19, 20, 21 from foot. The formation of a backwater depends essentially on the non-riegligibility of squares of velocities; and your p. 367, lines 1—4, and line 17 from foot, are not right.
{< If you come to the R S. Library Committee on Thursday we may come to agreement on some of these questions."
Although the main purpose in Kelvin's papers of 1887 was not attained, his special solution for a disturbed vorticity in case (i) is not without interest. The general dynamical equation for the vorticity in two dimensions is
D£_dr     dt      d?_   -                       m
~  + u+v~v * .....................(i)
where v (=plp') is the kinematic viscosity and V2 = d*/dix;* + d*ldyz. In this hydrodynamical equation £ is itself a feature of the motion, being connected with the velocities u, v by the relation
i    dv\                                   . .
I    dosj'..............................
while u, v themselves satisfy the " equation of continuity"
dx    dy
* I think Kelvin did not understand that the analogous elastic problem referred to is that of a thin plate.    See words following equation (5) of my paper.. I dipped a disturbing rod an inch or two down into the water and immediately the torque increased largely. Smooth regime could only be